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At one point, I remember being excited by seeing the website Encyclopedia of Combinatorial Structures as an extension of Sloane's Online Integer Sequence Database site. Unfortunately, the site (ECS) has already become largely defunct.

I would very much like to see a searchable database of combinatorial objects online -- ideally, not just as a site with pictures or counts of these structures, but a database with downloadable files in SOME format containing usable, interpretable copies of these enumerated sets (a la the ARG database for labeled and unlabeled graphs.)

Broadly requested, does anyone know of such a site or sites? In particular, I am asking as part of a search for lists of polyominos stored as matrices (rather than just displayed as a picture), but I'd be curious if such an encyclopaedic site (or sites) exists -- and if not in general, then I would love to be pointed to a few databases of specialized combinatorial structures.

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    $\begingroup$ I'm not sure that it is exactly what you're looking for, but findstat.org is another nice combinatorial database. $\endgroup$
    – Sam Hopkins
    Commented Nov 13, 2014 at 4:06
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    $\begingroup$ Does the ATLAS of finite simple groups count? $\endgroup$
    – Noam D. Elkies
    Commented Nov 13, 2014 at 4:28
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    $\begingroup$ A very similar topic was discussed by Bridget Tenner at Richard Stanley's 70th birthday conference. She refers to this sort of thing as a "fingerprint database." math.mit.edu/stanley70/Site/Slides/Tenner.pdf You might want to contact Tenner to see if there have been any recent developments. $\endgroup$
    – Timothy Chow
    Commented Nov 13, 2014 at 16:27
  • $\begingroup$ I misread the post title as meaning databases seen as an interesting pure combinatorial structure, which they are (as sets of interrelated relations) ! $\endgroup$
    – Claude Chaunier
    Commented Jan 26 at 12:58

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Not related to polyominos, but I like House of Graphs: https://houseofgraphs.org

It allows you to search by graph6 code, so if you find an interesting graph, you can check whether that graph has arisen in other contexts.

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  1. La Jolla Covering repository: https://www.dmgordon.org/cover/
  2. Information system on graph classes and their inclusions: http://www.graphclasses.org
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To expand Thomas Kalinowski's answer graphclasses.org can be used in sage http://sagemath.org/

Download the latest database and do custom queries in python, possibly adding inclusions, new classes, etc.

Check sage.graphs.isgci.graph_classes

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How about FindStat? It is a database of combinatorial statistics, i.e. maps from combinatorial objects to integers.

Another page I just found has a database with posets, which is quite nice.

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    $\begingroup$ FindStat was mentioned by Sam Hopkins in a comment. $\endgroup$
    – Gerry Myerson
    Commented Nov 13, 2014 at 11:09
  • $\begingroup$ Oh, I missed that. $\endgroup$
    – Per Alexandersson
    Commented Nov 13, 2014 at 11:32
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There is a database of linear block codes and quantum codes at http://www.codetables.de

There's a design database at http://designtheory.org/database/

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A similar question was asked here. At the time I suggested Donald Knuth's Stanford GraphBase: A Platform for Combinatorial Computing (1994, 2009) and the accompanying website.

See also The Stony Brook Algorithm Repository.

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Andries Brouwer's collection of strongly regular graphs: http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html

Eric Moorhouse's collections of finite projective planes and generalized polygons: http://ericmoorhouse.org/pub/planes/, http://ericmoorhouse.org/pub/genpoly/

Gordon Royle's collection of combinatorial objects: http://staffhome.ecm.uwa.edu.au/~00013890/data.html

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The Combinatorial Object Server (CoS) can generate perms/combs, sets/multisets, partitions, DeBuijn seqs/Lyndon words/necklaces, graphs, and a few others, all through a web interface. Unfortunately, it only typically works well for queries that have relatively few results. (It will truncate the responses to 1000 objects)

http://combos.org

(Edit: referring back to your original specific request, it can also produce all polyominoes of fixed size or all polyomino covering of a user-specified shape, and several others, and output as either text or images)

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  • $\begingroup$ The original COS website is broken since many years, but the new one on combos.org is fully functional and brings back much of the original content. $\endgroup$
    – Torsten Mütze
    Commented May 3, 2019 at 14:16
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Yoshitake Matsumoto maintains some databases of matroids. Gordon Royle also has a database of matroids that might contain some data not in Matsumoto's database, but I can't seem to find it on his webpage. The URL in his paper with Mayhew does not seem to work any more. You might have to contact Royle or Mayhew.

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Encyclopedia of Combinatorial Polytope Sequences by Stefan Forcey

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SMAPO is a library of linear descriptions of low-dimensional 0/1-polytopes connected with small instances of combinatorial optimization problems:

http://www.iwr.uni-heidelberg.de/groups/comopt/software/SMAPO/

(I quoted the description from the website.)

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